Magnetoelectric coupling in multiferroics probed by optical second harmonic generation

Magnetoelectric coupling, as a fundamental physical nature and with the potential to add functionality to devices while also reducing energy consumption, has been challenging to be probed in freestanding membranes or two-dimensional materials due to their instability and fragility. In this paper, we report a magnetoelectric coupling probed by optical second harmonic generation with external magnetic field, and show the manipulation of the ferroelectric and antiferromagnetic orders by the magnetic and thermal fields in BiFeO3 films epitaxially grown on the substrates and in the freestanding ones. Here we define an optical magnetoelectric-coupling constant, denoting the ability of controlling light-induced nonlinear polarization by the magnetic field, and found the magnetoelectric-coupling was suppressed by strain releasing but remain robust against thermal fluctuation for freestanding BiFeO3.

1. The symmetry of film form BiFeO3 is not addressed. Is that the rhombohedral R3c or tetragonal P4mm? Although all can be related to the pseudo-cubic structure, it is necessary to clarify this since different polymorphs of BFO have been reported in thickness/substrate dependent cases. 2. The cell evolution indicates that the lattice mismatch between the substrate and BFO introduce tensile stress to BFO since the dimension along the c-axis become smaller in freestanding state, says from 4.061 to 3.979 Å? Is this for the unit cell parameter c of the P4mm symmetry? Or the d value for certain (hkl)? 3. As far as I understood, structurally, the freestanding and strained BFO should be considered as different phases due to their large dimensional difference, since the magnetostriction-polarization coupling is commonly observed in related system, such as in Mn2MnWO6 (Nat. Commun. , 2017, 8, 2037. So the multiple coupling effect cannot be simply counted when spin-lattice-phonon-dipole are got involved. So it could be a plus to measure any samples before and after removal of the intermediate Sr-Al-O layer for comparison. 4. The SHG was detected by reflection mode. In the strained BFO film, the out-plane stress reaches maximum in the BFO and sbustrate interface, being significantly faded with growing film thickness. My question is, for the ~50 nm thickness film, what is the stress effect on the upper surface compared with the freestanding one? So it is better to evaluate strain effect on surfaces of both strained and freestanding (top and bottom) ones by the lattice dimension. Presumably, the thicker the film, the smaller the lattice dimension of the outer-plane, the limit should be equal to that of the freestanding case. 5. Is the freestanding film totally relaxed? Any SHG difference measured on the top and bottom (the side epitaxially grown on substrate) surfaces? 6. The SHG-magnetism-polarization-lattice coupling is rather complicated. The authors are brave enough to touch this. It will be better to calculate the spontaneous polarization according the cell-parameter-based cif files by either first-principles calculations or pointcharge model, so that one can tell the structurally dipole contribution. 7. The properties of these kind of films are very sensitive to defect, such as oxygen vacancy. Any characteristics on this -says how reproducible of the samples from different batches?

Title: Magnetoelectric Coupling in Multiferroics Probed by Optical Second Harmonic Generation
We thank all the reviewers for positive recommendation and valuable comments regarding our research paper. Each of your insights have served to strengthen our manuscript. We have carefully revised the manuscript according to your constructive suggestions. Provided below is our detailed response to each comment raised.
Reviewer #1: Comment. The paper reports important phenomena related to magneto-electric coupling in a ferroelectric material system by probing magnetic field dependence of the second harmonic generation. The study presents convincing results, but the authors need to discuss the possibility of magneto-striction in addition to magnetoelectric effects influencing the SHG. After adding this discussion the paper would be stronger.

Response.
We greatly appreciate the positive comments from this referee. We thank the reviewer for the constructive suggestions that are important for the improvement of the manuscript. We have addressed the reviewer's suggestion in the following and in the revised manuscript.
In the revised manuscript on page 3-4, we have added the relevant description as: "It has been found that the ferroelectric polarization in the BFO films directly coupled with the non-collinear G-type antiferromagnetic as well as the weak ferromagnetic moment driven by the Dzyaloshinskii-Moriya (DM) interaction, which arises from spin-orbital coupling in antisymmetric systems 17,28,29,30,31 ".
On page 5, we have rewritten the relevant description as: "The magnetoelectric coupling in BFO was induced by its intrinsic and significant spin-orbital coupling.
The antiferromagnetic order would introduce additional electric polarization via spinorbital coupling directly, which is well-known as magneto-striction phenomenon 33,34 . Furthermore, the coupling between the ferroelectric order and the non-collinear G-type antiferromagnetic order, which is induced by the DM interaction, offers a more complex way to manipulate the electric property and magnetic property by each other 35,36 . That is, the reduction of the ferroelectric order contributed to the SHG ( Fig.   2i-k) can be attributed to the orientation varying of the electric polarization 37      freestanding BFO) and the magnetic field-dependent SHG signal contributed by antiferromagnetic order for BFO/STO (and BFO/SAO/STO), and found they behaved similarly with those in Fig. S6 and S11a, respectively, and further systematic study is planned. We sincerely appreciate the reviewer for the inspiration!  The following quoted sentences have been added on page 6 in the revised manuscript: "The magnetic field-dependent SHG signal (extracted from Fig. 2h-k) contributed by antiferromagnetic order and ferroelectric order also exhibits an upset down parabolic behavior (Fig. S11)." In addition, we have preliminarily found the magnetic field-dependent SHG signal contributed by ferroelectric order for BFO/STO (and BFO/SAO/STO) seems similar with that in S11b, and further systematic study is planned. We deeply appreciate the reviewer's inspiration.   (Fig. S1a), RSM (Fig. S1g-l), fast fourier transform (FFT) patterns (Fig. S3), and RA-SHG (Fig. 3f-g)  In the revised manuscript, the following quoted sentences have been added on page 3: "From the measurements of XRD (Fig. S1a), RSM (Fig. S1g-l), FFT patterns ( Fig. S3), and RA-SHG, it can be concluded that: with the thickness of 47 nm, the stress on the BFO films from the substrates was well kept, so that the epitaxial BFO films on STO and those on SAO/STO were in a tetragonal-like (T-like) phase and most likely with a P4mm space group, consistent with our previous study 19

Response 3. Thanks for the reviewer's valuable suggestions on our manuscript. By following the reviewer's inspiring suggestion, we have revised the relevant discussion
in the revised manuscript on page 5 as: "The magnetoelectric coupling in BFO was induced by its intrinsic and significant spin-orbital coupling. The antiferromagnetic order would introduce additional electric polarization via spin-orbital coupling directly, which is well-known as magneto-striction phenomenon 33,34 . Furthermore, the coupling between the ferroelectric order and the non-collinear G-type antiferromagnetic order, which is induced by the DM interaction, offers a more complex way to manipulate the electric property and magnetic property by each other 35,36 ." We have added three new references paper as Ref. 33 We performed the XRD, RSM (Fig. S1) (Fig. S1a), RSM ( Fig. S1g-l), FFT patterns (Fig. S3), and RA-SHG, it can be concluded that: with the thickness of 47 nm, the stress on the BFO films from the substrates was well kept, so that the epitaxial BFO films on STO and those on SAO/STO were in a tetragonal-like (T-like) phase and most likely with a P4mm space group, consistent with our previous study 19 , while it was also feasible to obtain the large-scale freestanding BFO film which stayed in a rhombohedral-like (R-like) phase and with an R3c space group." We have added one new reference paper as Ref. 19 Figure   R3 and below: Figure R3:  Figure R4 and below: List of changes to the manuscript 1. In the revised manuscript on page 3-4, we have added the relevant description as: "It has been found that the ferroelectric polarization in the BFO films directly coupled with the non-collinear G-type antiferromagnetic as well as the weak ferromagnetic moment driven by the Dzyaloshinskii-Moriya (DM) interaction, which arises from spin-orbital coupling in antisymmetric systems 17,28,29,30,31 ".

2.
On page 5, we have rewritten the relevant description as: "The magnetoelectric coupling in BFO was induced by its intrinsic and significant spin-orbital coupling.
The antiferromagnetic order would introduce additional electric polarization via spinorbital coupling directly, which is well-known as magneto-striction phenomenon 33,34 .
Furthermore, the coupling between the ferroelectric order and the non-collinear Gtype antiferromagnetic order, which is induced by the DM interaction, offers a more complex way to manipulate the electric property and magnetic property by each other 35,36 . That is, the reduction of the ferroelectric order contributed to the SHG ( Fig.   2i-k) can be attributed to the orientation varying of the electric polarization 37  3. we have also addressed the phase of BFO films and added the following sentences on page 3 of the revised manuscript: "From the measurements of XRD (Fig. S1a), RSM ( Fig. S1g-l), FFT patterns (Fig. S3), and RA-SHG, it can be concluded that: with the thickness of 47 nm, the stress on the BFO films from the substrates was well kept, so that the epitaxial BFO films on STO and those on SAO/STO were in a tetragonal-like (T-like) phase and most likely with a P4mm space group, consistent with our previous study 19  between the polarization strength with respect to the offset between the Fe cation and the four surrounding Bi cations was adopted to obtain the polarization semiquantitatively for BFO/STO, BFO/SAO/STO, and freestanding BFO films. The profile of polarization (Fig. 1j), also denoted by the yellow vectors in Fig. 1g-   Reviewer #1: Comment. The authors provided satisfactory responses to all of the comments presented by the reviewers. They also modified the paper accordingly. I recommend the publication of the paper and I trust the authors will carefully proofread and crosscheck the information in the text.

Response.
We greatly appreciate the positive comments from this referee. We have revised the corresponding content and format of the manuscript according to the Author Checklist.